There are mostly four kinds of coins in circulation in the U.S: 1 cent, 5 cents, 10 cents, and 25 cents. But is it the most efficient way to give back change?

This Science News article says that a computer scientist has found an answer.

In the current issue of the Mathematical Intelligencer, [Jeffrey Shallit of the University of Waterloo] contends that "what the U.S. needs is an 18-cent piece."

In finding coin denominations that minimize the average cost of making change, Shallit assumed that every amount of change between 0 and 99 cents is equally likely. For the current four-denomination system, he found that, on average, a change-maker must return 4.70 coins with every transaction.

He discovered two sets of four denominations that minimize the transaction cost. The combination of 1 cent, 5 cents, 18 cents, and 25 cents requires only 3.89 coins in change per transaction, as does the combination of 1 cent, 5 cents, 18 cents, and 29 cents.

Of course, it would be more difficult for cashiers to give you back your change. So, instead of replacing one of the existing four coins, Shallit played with the hypothesis of the addition of a fith coin.

It turns out that the greatest improvement in change-making efficiency would occur with the addition of a 32-cent coin. This reduces the average cost to 3.46 coins per transaction.

Is this a crazy idea? Other coins existed in the past.

The United States has experimented briefly with extra coin denominations. At one time or another in the distant past, the U.S. mint issued half-cent, two-cent, three-cent, and 20-cent piecesóbut it never produced an 18-cent coin.

Obviously, this optimization of change can also be done in other countries.

[In Canada,] Shallit's calculations show that the average cost of making change would fall from 5.90 to 4.58 coins per transaction with the addition of an 83-cent coin.

Similarly, the new Euro system introduced by the European Union would benefit from the addition of a 1.33- or 1.37-Euro coin.